Simulation #1

Data structure: \(O = (W, A, Z, Y)\)

  • U - exogenous variables
  • W - baseline covariate that is a measure of body condition
  • A - treatment level based on W, continuous between 0 and 5
  • Z - intermediate curve based on W and A
  • Y - outcome, indicator of an event ?

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
    • \(U_Z \sim Uniform(min = 0, max = 1)\)
    • \(U_Y \sim Uniform(min = 0, max = 1)\)
  • Structural equations F and endogenous variables:
    • \(W = U_W\)
    • \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
    • \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
    • \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
##        W                   A                Z                Y         
##  Min.   :-3.829933   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:-0.662628   1st Qu.:0.6306   1st Qu.:0.0000   1st Qu.:0.0000  
##  Median :-0.004864   Median :2.0093   Median :0.0000   Median :1.0000  
##  Mean   :-0.002210   Mean   :2.1248   Mean   :0.4932   Mean   :0.5977  
##  3rd Qu.: 0.660425   3rd Qu.:3.4035   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   : 4.311039   Max.   :5.0000   Max.   :1.0000   Max.   :1.0000
## Summary of A given W < -1:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   1.407   2.798   2.689   4.029   5.000
## Summary of A given -1 < W <= 0:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0000  0.9264  2.2402  2.3016  3.5982  5.0000
## Summary of A given 0 < W <= 1:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0000  0.4099  1.7462  1.9385  3.1730  5.0000
## Summary of A given 1 < W:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   0.000   1.255   1.574   2.746   5.000

## 
## Call:
## glm(formula = Y ~ W + A + W * A + Z, family = binomial, data = obs)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.0483  -0.0492   0.0004   0.0467   3.7830  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -8.6327     0.2789 -30.952   <2e-16 ***
## W             0.6968     0.2233   3.121   0.0018 ** 
## A             5.3060     0.1566  33.877   <2e-16 ***
## Z             1.1786     0.1234   9.551   <2e-16 ***
## W:A          -0.2709     0.1444  -1.876   0.0606 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 13478.7  on 9999  degrees of freedom
## Residual deviance:  2407.3  on 9995  degrees of freedom
## AIC: 2417.3
## 
## Number of Fisher Scoring iterations: 9
## [1] "    MSE: 365.4844, AUC: 0.9908"

n = 50

CV HAL

results

## CV selected lambda (from one sample): 0.00990300430080305

1000 repetition

## The average of CV selected lambdas (from 1000 sample): 0.0177983801957674 The average of CV selected lambdas (from 1000 sample): 0.0177862720162932 The average of CV selected lambdas (from 1000 sample): 0.0178028065364971 The average of CV selected lambdas (from 1000 sample): 0.0177907302552206
## z=1:

## z=0:

Globally Undersmoothed HAL

results

## Undersmoothed lambda: 0.00037559027270435
##  which is 0.0379269019073225 * lambda_CV

1000 repetition

## The average of unsersmoothed lambda (from 1000 sample): 0.000581076546605094
##  which is 0.0329348525092873 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580903922299673
##  which is 0.0329303125559549 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580778134680317
##  which is 0.0329164209081906 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580426573286071
##  which is 0.0329164209081906 * the average of 1000 lambda_CV
## z=1:

## z=0:

Oevr a grid of lambda scalers

## TableGrob (7 x 4) "arrange": 7 grobs
##   z     cells    name                 grob
## 1 1 (2-3,2-3) arrange       gtable[layout]
## 2 2 (4-5,1-2) arrange       gtable[layout]
## 3 3 (4-5,3-4) arrange       gtable[layout]
## 4 4 (6-7,1-2) arrange       gtable[layout]
## 5 5 (6-7,3-4) arrange       gtable[layout]
## 6 6 (3-3,4-4) arrange    gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.1916]

## TableGrob (7 x 4) "arrange": 7 grobs
##   z     cells    name                 grob
## 1 1 (2-3,2-3) arrange       gtable[layout]
## 2 2 (4-5,1-2) arrange       gtable[layout]
## 3 3 (4-5,3-4) arrange       gtable[layout]
## 4 4 (6-7,1-2) arrange       gtable[layout]
## 5 5 (6-7,3-4) arrange       gtable[layout]
## 6 6 (3-3,4-4) arrange    gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.2192]

## TableGrob (7 x 4) "arrange": 7 grobs
##   z     cells    name                 grob
## 1 1 (2-3,2-3) arrange       gtable[layout]
## 2 2 (4-5,1-2) arrange       gtable[layout]
## 3 3 (4-5,3-4) arrange       gtable[layout]
## 4 4 (6-7,1-2) arrange       gtable[layout]
## 5 5 (6-7,3-4) arrange       gtable[layout]
## 6 6 (3-3,4-4) arrange    gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.2468]